Question: Solve for $x$ : $ 6|x + 6| + 5 = -3|x + 6| + 7 $
Answer: Add $ {3|x + 6|} $ to both sides: $ \begin{eqnarray} 6|x + 6| + 5 &=& -3|x + 6| + 7 \\ \\ { + 3|x + 6|} && { + 3|x + 6|} \\ \\ 9|x + 6| + 5 &=& 7 \end{eqnarray} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} 9|x + 6| + 5 &=& 7 \\ \\ { - 5} &=& { - 5} \\ \\ 9|x + 6| &=& 2 \end{eqnarray} $ Divide both sides by ${9}$ $ \dfrac{9|x + 6|} {{9}} = \dfrac{2} {{9}} $ Simplify: $ |x + 6| = \dfrac{2}{9}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 6 = -\dfrac{2}{9} $ or $ x + 6 = \dfrac{2}{9} $ Solve for the solution where $x + 6$ is negative: $ x + 6 = -\dfrac{2}{9} $ Subtract ${6}$ from both sides: $ \begin{eqnarray} x + 6 &=& -\dfrac{2}{9} \\ \\ {- 6} && {- 6} \\ \\ x &=& -\dfrac{2}{9} - 6 \end{eqnarray} $ Change the ${ - 6}$ to an equivalent fraction with a denominator of $9$ $ x = - \dfrac{2}{9} {- \dfrac{54}{9}} $ $ x = -\dfrac{56}{9} $ Then calculate the solution where $x + 6$ is positive: $ x + 6 = \dfrac{2}{9} $ Subtract ${6}$ from both sides: $ \begin{eqnarray} x + 6 &=& \dfrac{2}{9} \\ \\ {- 6} && {- 6} \\ \\ x &=& \dfrac{2}{9} - 6 \end{eqnarray} $ Change the ${ - 6}$ to an equivalent fraction with a denominator of $9$ $ x = \dfrac{2}{9} {- \dfrac{54}{9}} $ $ x = -\dfrac{52}{9} $ Thus, the correct answer is $x = -\dfrac{56}{9} $ or $x = -\dfrac{52}{9} $.